Децентрализованная оптимизация с аффинными ограничениями / Decentralized optimization with affine constraints тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Ярмошик Демьян Валерьевич

• Ярмошик Демьян Валерьевич
• кандидат наук
• 2024

Введение диссертации (часть автореферата) на тему «Децентрализованная оптимизация с аффинными ограничениями / Decentralized optimization with affine constraints»

Introduction

1.1 Decentralized Optimization and Affine Constraints

Decentralized optimization is a significant branch of numerical optimization research. Its development began with the consideration of distributed optimization algorithms [8,72]. Later, the notion of decentralized optimization over communication graphs was introduced [53], combining ideas from distributed optimization and consensus algorithms. We provide a more comprehensive literature review in dedicated sections of the following chapters.

Essentially, decentralized optimization is concerned with finding a solution to an optimization problem, di erent parts of which are kept privately at di erent computing nodes. The computing nodes are connected through a communication graph, and the atomic communication operation allows the exchange of information (i.e., numeric arrays) with immediate neighbors only.

Our thesis focuses on decentralized optimization with a ne constraints. A ne constraints are often artificially introduced to solve problems of form ^^ f (x) ^ minx, where f are stored at di erent computing nodes, with the consensus-based reformulation: ^ fi(xi) ^ minx s.t. xi = ... = xn. Since the resulting formulation has separable objective and the variables are only connected through the a ne constraint, such constraints are called coupled. Coupled constraints are also naturally arise in various sharing and resource allocation problems, e.g. in the optimal power flow problem. Sometimes, in addition to the consensus constraint xi = ... = xn each node has another restriction on its variable's value posed as AiXi = bi. Such constraints are called local.

In this thesis we propose new algorithms for decentralized optimization problems with a ne constraints under various assumptions. We provide convergence analysis of our algorithms and construct lower bounds to show their optimality. Numerical experiments are conducted to evaluate practical performance of our algorithms.

1.2 Scientific Novelty

We underline the three main contributions of the thesis:

• Analysis of the decentralized Extragradient method for convex smooth optimization with mixed a ne constraints

• Lower bounds and optimal first-order algorithms for strongly convex decentralized optimization with local a ne constraints on static and time-varying graphs

• Optimal first order decentralized algorithm for coupled constraints in strongly convex setup

Below we explain our contributions in more detail.

1

In Chapter 2 we consider our most general formulation of problem with a ne constraints, which include coupled and local constraints posed as equalities and inequalities. We apply Extragradient method to a decentralized-friendly reformulation of this problem and derive the convergence rates for function value and constraints deviation errors using parameters of the original problem's formulation. Here we assume static communication graph and smooth, convex, but not strongly convex objective function.

In Chapter 3 we focus on local a ne constraints. We adapt several di erent algorithmic approaches for general a ne-constrained optimization to the (consensus) decentralized optimization with local constraints. Utilizing special structure of our consensus-based reformulation, we separate spectral properties of the local constraint's matrix and the gossip matrix in the convergence rates for smooth and strongly convex objectives. These results are extended in Chapter 4 to the time-varying communication graphs, and corresponding lower bounds are also established for both static and time-varying cases.

In Chapter 5 we propose an algorithm with accelerated linear convergence rate for coupled constraints. Due to a degeneracy that appear in decentralized-friendly reformulations of problems with coupled constraints, it is not straightforward to obtain a linearly convergent algorithm. Using augmented Lagrangian technique we develop a first decentralized gradient-descent-like algorithm with linear convergence for coupled constraints. Utilization of Nesterov's and Chebyshev's accelerations allows us to obtain the optimal convergence rate. The lower bound that matches the convergence rate is constructed in this chapter.

The papers, on which the present thesis is based, were prepared with co-authors. The table below specifies the amount of personal contribution of the author in each paper.

Table 1.1: Personal contribution of the author in theoretical results (Theory), text writing (Text) and

numerical experiments (Experiments) for each chapter

Chapter Base paper Theory Text Experiments

Chapter 2 [15] 70% 80% 100%

Chapter 3 [17] 35% 50% 50%

Chapter 4 [81] 80% 90% 100%

Chapter 5 [14] 35% 65% 0%

1.3 Presentations and Validation of Research Results

The results were presented at the following conferences

1. Decentralized Convex Optimization under A ne Constraints for Power Systems Control. International Conference on Mathematical Optimization Theory and Operations Research, Petrozavodsk, Karelia, Russia, July 2-6, 2022.

2. Decentralized Strongly-Convex Optimization with A ne Constraints: Primal and Dual Approaches. XIII International Conference on Optimization and Applications. Petrovac, Montenegro, September 26, 2022 (online).

1.4 Publications

The thesis is based on the following publications

Published papers.

1. Yarmoshik, D., Rogozin, A., Khamisov, O.O., Dvurechensky, P., Gasnikov, A. (2022, June). Decentralized Convex Optimization Under A ne Constraints for Power Systems Control. In: Pardalos, P., Khachay, M., Mazalov, V. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2022. Lecture Notes in Computer Science, vol 13367. Springer, Cham. [15]

2. Rogozin, A., Yarmoshik, D., Kopylova, K., and Gasnikov, A. (2023, January). Decentralized Strongly-Convex Optimization with A ne Constraints: Primal and Dual Approaches. In Advances in Optimization and Applications: 13th International Conference, OPTIMA 2022, Petrovac, Montenegro, September 26-30, 2022, Revised Selected Papers (pp. 93-105). Cham: Sprijnger Nature Switzerland. [17]

3. Yarmoshik, D., Rogozin, A., Gasnikov, A. (2023, October). Decentralized optimization with ane constraints over time-varying networks. Comput Manag Sci 21, 10. [81]

1.5 Structure of the Thesis

The thesis consists of an introduction, four main chapters, list of references and a conclusion.

Chapter

Mixed Affine Constraints. Extragradient Method

Summary

Modern power systems are now in continuous process of massive changes. Increased penetration of distributed generation, usage of energy storage and controllable demand require introduction of a new control paradigm that does not rely on massive information exchange required by centralized approaches. Distributed algorithms can rely only on limited information from neighbours to obtain an optimal solution for various optimization problems, such as optimal power flow, unit commitment etc.

As a generalization of these problems we consider the problem of decentralized minimization of the smooth and convex partially separable function f = ^k=i fk (xk, x) under the coupled ^k=i (Akxk — bk) < 0 and the shared Ax — b < 0 a ne constraints, where the information about Ak and bk is only available for the k-th node of the computational network.

One way to handle the coupled constraints in a distributed manner is to rewrite them in a distributed-friendly form using the Laplace matrix of the communication graph and auxiliary variables (Khamisov, CDC, 2017). Instead of using this method we reformulate the constrained optimization problem as a saddle point problem (SPP) and utilize the consensus constraint technique to make it distributed-friendly. Then we provide a complexity analysis for state-of-the-art SPP solving algorithms applied to this SPP.

Заключение диссертации по теме «Другие cпециальности», Ярмошик Демьян Валерьевич

Conclusion

This thesis featured decentralized optimization algorithms designed for problems with a ne constraints. We covered di erent variants of problem formulations. For strongly convex objectives we obtained optimal algorithms and corresponding lower bounds on the complexity of first-order algorithms. All proposed algorithms come with provable convergence rates showing the dependency on problem's parameters.

The potential for applying our algorithms in practice was demonstrated with numerical experiments on real-world problems such as optimal power flow and vertical federated learning. However, we are not yet certain, whether the decentralized setting is useful in practice, or if another distributed protocols with higher level of coordination (e.g. centralized) can be applied instead, facilitating more e cient information exchange while preserving privacy and failure-tolerance, and avoiding communication bottlenecks.

Several open questions remain for future work. First, linearly convergent algorithms for coupled constraints on time-varying networks are unknown. Second, it is desirable to obtain first-order algorithms for more general problem formulations, e.g. including bounded domains of objective functions Xj £ Qj and coupled inequalities with general convex functions ^j gj(xj) < 0. Third, an important for practical applications and not yet implemented feature is the adaptivity of the algorithms. Even incorporation of backtracking adaptive stepsize will significantly reduce the burden of parameter tuning.

Список литературы диссертационного исследования кандидат наук Ярмошик Демьян Валерьевич, 2024 год

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